The Trace Formula for Noncompact Quotient

1. In [12] and [13] Selberg introduced a trace formula for a compact, locally symmetric space of negative curvature. There is a natural algebra of operators on any such space which commute with the Laplacian. The Selberg trace formula gives the trace of these operators. Selberg also pointed out the importance of deriving such a formula when the symmetric space is assumed only to have finite volume. Then the Laplace operator will have continuous as well as discrete spectrum; it is the trace of the restriction of the operator to the discrete spectrum that is sought. Selberg gave such a formula for the quotient of the upper half plane by SL(2, Z). (See also [6] and [8].) Selberg also suggested how to extend the formula to any locally symmetric space of rank 1. Spaces of rank 1 are the easiest noncompact ones to handle for they can be compactified in a natural way by adding a finite number of points. I have recently obtained a trace formula for spaces of higher rank. In this article I shall illustrate the formula by looking at a typical example.